Target signal extraction has a great potential for applications. To solve the problem of error extraction of target signals in the current constrained independent component analysis (cICA) method, an enhanced independent component analysis with reference (EICAR) method is proposed. The new algorithm establishes a unified cost function, which combines the negative entropy contrast function and the distance metric function. The EICAR method transforms the constrained optimization problem into unconstrained optimization problem to overcome the problem of threshold setting of distance metric function in constrained optimization problem. The theoretical analysis and simulation experiment show that the proposed EICAR algorithm overcomes the problem of the error extraction of the existing algorithm and improves the reliability of the target signal extraction.
Target signal extraction is used to extract unknown source signals from multiple linear mixed signals, which has found a wide range of applications. Especially in the case of the complex electromagnetic environment, a substantial number of electromagnetic signals are interwoven together to interfere with each other [
A current trend in target signal separation is the independent component analysis (ICA) approach, the core idea of which is to minimize the statistical relationship between all the signal sources [
Although the ICA method can separate the mixed signals to some extent, the signal sorting order separated by ICA is only related to the nonGaussian of the source signal [
In many practical applications, some characteristics of the target signal, such as the carrier frequency, modulation mode, and other prior information are known, which can be used for target signal extraction. If there is a frequency aliasing between the signals, the signal cannot be separated by the traditional filtering method. In this case, the constrained ICA (cICA) algorithm [
However, in the process of optimization for the cICA algorithm, we need to set threshold parameters to distinguish target signals from other signals, which increases computation complexity and storage space and converges slowly. In some cases, the cICA algorithm cannot be converged [
Recently, Shi et al. proposed a new model of ICA with reference signal (ICAR), where an adaptive weighted summation method is introduced to solve the multiobjective optimization problem with a new fixedpoint learning algorithm [
Compared with the cICA algorithm or ICAR, the proposed enhanced ICA with reference (EICAR) directly contains the prior information into the ICA framework. By combining with the negative entropy contrast function and target signal distance metric function, the EICAR establishes a unified cost function so that the constrained optimization problem is transformed into an unconstrained optimization to overcome the problem of the threshold setting problem of distance metric function for cICA.
In the enhanced ICA with reference (EICAR) proposed in this paper, a priori information is directly contained in the ICA framework combined with the negative entropy contrast function and target signal distance metric function.
The EICAR puts forward four kinds of cost function to convert the constrained optimization problem into unconstrained optimization problem. By deductive analysis of the similarity of the four cost functions, EICAR establishes a unified optimization model, in which the model weight parameter is determined to meet four kinds of cost function at the same time. It not only overcomes the difficulty of setting the threshold for distance measurement function but also solves the difficulty of setting weight parameter.
In practice, the reference signal can be obtained in advance. Under the counter condition, the interference signal is not completely consistent with the target signal in frequency and may only overlap partially. Even if it is completely in the same frequency, the modulation mode of target signal and interference signal will be different. Moreover, interference signals are usually strong noises or direct background music and other unrelated signals, which are significantly different from target signals. In addition, the transmission time and mode of target signals have certain rules, while interference signals generally lack such rules. In general, continuous interference is adopted, or the same frequency interference is sent after the detection of target signals, which has obvious lag in time. We can predict in advance the precise frequency, modulation mode, and even the law of signal transmission of the target transmitter, which can be used as the basis for designing reference signals. Accordingly, interference signal does not have these characteristics.
Of course, the reference signal we designed is only an approximate version of the “expected” reference signal, but this does not affect the validity of the results. Because the reference signal is not required to be infinitely close to the actual target signal, only the distance measurement function with the target signal is the minimum. Since the reference signal is designed based on some features of the target signal, it is obvious that its distance measurement function with the target signal is smaller than that with other interference signals.
The research in this paper focuses on target signal extraction mainly targeted at nonunderdetermined system. In practice, some hybrid systems are underdetermined. In terms of the underdetermined system, scholars such as Woo et al. have conducted some fruitful researches [
The rest of this paper is organized as follows. In Section
The
The signal separation model can simply be described as follows: for the observed signal
In order to reduce the computation in the iteration process, we need to remove the correlation between the observed signals by whitening the data with
It is assumed that the target signal is
Based on the prior information of the target signal, the reference signal
As a result, the distance between the target signal
For equation (
It is very difficult to calculate the negative entropy, so the nonquadratic function is often used to approximate the negative entropy [
Accordingly,
To extract the target signal
The cICA algorithm makes use of maximization negative entropy method to solve the target signal
It is difficult to set the threshold parameter
To solve the problem of setting threshold parameters
With the combination of (
In order to describe the two optimization problems in a unified way, the maximization and minimization can be transformed into each other.
Direction 1: combining the distance metric function and the negative entropy contrast function, we reduce the constrained conditions and establish two forms of cost functions
Since the selection of
Direction 2: combining the distance metric function and the negative entropy contrast function, we reduce the constrained conditions and establish two forms of cost functions
The scheme of (
According to the comprehensive analysis of equations (
Now, we will try to find out the common features of equations (
The gradient of the cost function for the four EICAR schemes
the gradient of the cost function for
They can be uniformly expressed as
The gradient of the cost function for
They can be uniformly expressed as
So they can be expressed in a similar form:
It can be obtained by using Theorem 1 that we can take a particular
The gradient of
For
From equations (
Thus, the gradientbased learning algorithm is shown in the following equation:
Input: The observation signal
Output: The target signal
Step 1: Preprocessing: whiten the observational signal
Step 2: Initialize:
Step 2.1: Determines the initial separation vector
Step 2.2: Determines the initial estimation signal
Step 2.3: Determine the initial parameters:
Step 3: Iterations:
Step 3.1: According to equation (
Step 3.2: Normalize
Step 3.3: Update
Step 3.4: Update the parameters
Step 3.6: Update the difference
Step 3.7: Compare
Step 4: Output results:
In the experiment, 10 groups of analog signals with different systems were selected. A total of 1000 experiments were conducted. One group of signals, in which the frequency of each signal was close to each other, is shown as follows: the source signals
For the convenience of displaying signals, we take the frequency as
Waveform diagram of source signals.
Spectra of source signals. (a)
An
Waveform diagram of mixed signals.
Spectra of mixed signals. (a)
For the target signal we need, this prior information is desirable. For example, in the case of 4 × 4 mixture, we need to analyze the number of the target signals. Take communication signals as an example, one of which is our normal communication signals and the other is antijamming signals and unintentional jamming signals. Then, only our normal communication signals are our target signals. The prior information of transmitter signal of our communication object can be known, which is sufficient to extract the target signal we need.
The reference signals should carry the prior information of the expected source signals with nonGauss characteristic. There are many kinds of reference signal design, and the most typical method is the pulse method. We select the pulse signals with the same frequency as the source signals as the reference signals.
In practice, for example, only one of the 4 signals need to be extracted, which means that only one signal is needed to extract the source signal.
In the simulation experiment, in order to compare the performance with cICA, we designed the reference signal for each signal and extracted each signal.
In order to compare the separation effect conveniently, we carry out the separation experiment by means of the EICAR method proposed in this paper and the cICA method.
Firstly, we use the method of cICA to carry out simulation experiments [
According to the experiment using the cICA method, different reference signals appear many times and the same target signal is extracted. For 1000 experiments with the cICA method, about 1/10 was erroneously extracted. Some results of these experiments are shown in Figures
The extraction effect 1 of the cICA method.
Spectra 1 of the cICA method. (a)
The extraction effect 2 of the cICA method.
Spectra 2 of the cICA method. (a)
The key to the cICA algorithm is the setting of the threshold
Then, we make use of the EICAR method proposed to carry out the experiment. The result of 1000 times shows that no extraction error occurred, as shown in Figures
The extraction effect of the EICAR method.
Spectra of the EICAR method. (a)
The experimental results of Figures
On this basis, we continue to study and compare the separation and extraction performance. First, we study the SNR of the extracted target signal, and the SNR of each target signal extracted is as follows:
The average SNR (dB) of the 1000 experiments.
The separate signal  FastICA  cICA  EICAR 


39.2  29.9  34.0 

38.4  28.2  34.5 

38.9  27.1  37.3 
Table
The corresponding run time is shown in Table
The average run time(s) of the 1000 experiments.
The separate signal  FastICA  cICA  EICAR 


0.328  0.176  0.152 

0.165  0.146  

0.181  0.163 
Table
Extracting the target signal accurately from the mixed signal is one of the difficulties in the field of signal processing. Based on the existing cICA algorithms, we propose an enhanced independent component analysis with reference (ICAR) to overcome the shortcomings of random and false extraction of separate signal sequence in the existing hybrid signal separation algorithm. By combining the negative entropy contrast function and the distance metric function of the target signal, we establish a unified cost function, which transforms the constrained optimization problem into an unconstrained optimization problem. The EICAR algorithm proposed in this paper not only overcomes the threshold setting problem of distance measurement function but also solves the problem of weight parameter setting. Theoretical analysis and simulation results show that the proposed ICAR algorithm outperforms the existing algorithms in extracting the target signal.
The data used to support the findings of this study are available from the corresponding author upon request.
The initial research of this paper was published in the Conference summary of the 5th International Conference on Information Science and Control Engineering (ICISCE) in 2018.
The authors declare that they have no conflicts of interest.
All the authors made theoretical and experimental verification and analysis of the content.
This paper has been supported by the National Natural Science Foundation of China (grant nos. 61602511 and 61401513).